Chemical Master versus Chemical Langevin for First-Order Reaction Networks

Markov jump processes are widely used to model interacting species in circumstances where discreteness and stochasticity are relevant. Such models have been particularly successful in computational cell biology, and in this case, the interactions are typically first-order. The Chemical Langevin Equation is a stochastic differential equation that can be regarded as an approximation to the underlying jump process. In particular, the Chemical Langevin Equation allows simulations to be performed more effectively. In this work, we obtain expressions for the first and second moments of the Chemical Langevin Equation for a generic first-order reaction network. Moreover, we show that these moments exactly match those of the underlying jump process. Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation. Our work assumes that a unique solution exists for the Chemical Langevin Equation. We also show that the moment matching result extends to the case where a gene regulation model of Raser and O’Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin equations. We finish with numerical experiments on a dimerization model that involves second order reactions, showing that the two regimes continue to give similar results.